15 21 23 triangle

Acute scalene triangle.

Sides: a = 15   b = 21   c = 23

Area: T = 153.7310893122
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 39.53662915601° = 39°32'11″ = 0.69900384618 rad
Angle ∠ B = β = 63.0243617313° = 63°1'25″ = 1.10999696286 rad
Angle ∠ C = γ = 77.44400911269° = 77°26'24″ = 1.35215845632 rad

Height: ha = 20.49774524162
Height: hb = 14.64110374402
Height: hc = 13.36879037497

Median: ma = 20.70662792408
Median: mb = 16.33224829711
Median: mc = 14.16986273153

Inradius: r = 5.2111216716
Circumradius: R = 11.78219519761

Vertex coordinates: A[23; 0] B[0; 0] C[6.80443478261; 13.36879037497]
Centroid: CG[9.93547826087; 4.45659679166]
Coordinates of the circumscribed circle: U[11.5; 2.5622107017]
Coordinates of the inscribed circle: I[8.5; 5.2111216716]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.464370844° = 140°27'49″ = 0.69900384618 rad
∠ B' = β' = 116.9766382687° = 116°58'35″ = 1.10999696286 rad
∠ C' = γ' = 102.5659908873° = 102°33'36″ = 1.35215845632 rad

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How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 15 ; ; b = 21 ; ; c = 23 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 15+21+23 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-15)(29.5-21)(29.5-23) } ; ; T = sqrt{ 23633.19 } = 153.73 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 153.73 }{ 15 } = 20.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 153.73 }{ 21 } = 14.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 153.73 }{ 23 } = 13.37 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 15**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 39° 32'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-15**2-23**2 }{ 2 * 15 * 23 } ) = 63° 1'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-15**2-21**2 }{ 2 * 21 * 15 } ) = 77° 26'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 153.73 }{ 29.5 } = 5.21 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 39° 32'11" } = 11.78 ; ;




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